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Upper bound of a volume exponent for directed polymers in a random environment. (English) Zbl 1041.60079

Summary: We consider the model of directed polymers in a random environment introduced by Petermann: the random walk is \(\mathbb R^d\)-valued and has independent \(\mathcal N(0,I_d)\)-increments, and the random medium is a stationary centered Gaussian process \((g(k,x), k\geqslant 1, x \in \mathbb R^d)\) with covariance matrix cov\((g(i,x),g(j,y))=\delta_{ij}\Gamma(x-y)\), where \(\Gamma\) is a bounded integrable function on \(\mathbb R^d\). For this model, we establish an upper bound of the volume exponent in all dimensions \(d\).

MSC:

60K37 Processes in random environments
60G42 Martingales with discrete parameter
82D30 Statistical mechanics of random media, disordered materials (including liquid crystals and spin glasses)
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